Jeffrey Chudakoff, Robert Cordiak and Nick GrattoThe Backbone of Mathematics

AwardBlue Ribbon

SchoolKenston High School

TeacherGreg Koltas

Selected Research"Neuropathic Pain Study after Spinal Cord Injury"

Selected ArtNumb

Selected Language"The Story of Kosan"

The mathematical concepts within the human vertebrae are
both abundant and complex, and their existence is vital to our
well-being. We find out just how structured our spine is within
Brianna Zahir's Numb. Upon reflection of "The Story of Kosan," we
discover the correlation between a sta and its trajectory. Aspects
of experimental design are examined and expanded on in Emerson
Thacker's spinal column research.

- Jeffrey Chudakoff, Robert Cordiak, Nick Gratto

Research Analysis

In his experiment, Emerson Thacker begins with four male rats,
which in this instance will be the samples. Each rat undergoes a
procedure known as the dorsal column crush, in which the rat's
spine is exposed at the C8 vertebrae after performing a T1
laminectomy. An incision is made on the vertebrae and a dorsal
crush column lesion is made, simulating the circumstances of a
patient who has incurred spinal cord injury (SCI).

Each of the four rats was then put through two behavioral tests
on a weekly basis to gauge spinal cord function. First, he tested
for allodynia which is a painful response to an event the subject
does not normally find painful. An example of this would be putting
on a shirt while sunburned - while the event would not normally
cause pain, this instance would be incredibly uncomfortable. In the
experiment, a filament of increasing size was mechanically inserted
into the hind paw of each rat until the rat pulled his paw away.
The rat was tested the week prior to the operation, and then tested
each week following the operation to measure the duration and
magnitude of spinal cord damage.

When mechanically stimulating the rats, Thacker utilized the Von
Frey monofilament set to test the effects of spinal cord damage.
The basic concept is this: when the tip of a fiber of given length
and diameter is pressed against the skin at a right angle, the
force of application increases as long as the probe is consistently
advanced, until the fiber bends. When the fiber bends, continued
advance creates more bend, but not more force of application. This
principle makes it possible for the researcher using a handheld
probe to apply a reproducible force, within a wide tolerance, to
the skin surface.

Traditionally, the force applied on the rat's skin can be
explained through the equation as stated in Newton's Second Law of
Motion: f = m(dv / dt) = ma. The filament is utilized to
apply the force onto the subject in a consistent, measurable
manner. The force at which the monofilament bends is directly
proportional to the diameter, but inversely proportional to its
length (though the filaments are all of constant length). The force
which the filament exerts upon the subject is represented by the
logarithmic function d = 4.0037 + .4581ln(f ), where d is
the diameter of the filament in micrometers and f is the
force in gram-force. (The gram-force is the force exerted on a 1
gram mass by the Earth's gravitational acceleration; 9.8
m/sec2). Therefore, based on the conversion between
newtons and gram-force (101.9716gf = 1N),
1.5N(101.9716) = 152.9575gf. By these calculations, we can
use the previous equation to determine the length of a filament
required to puncture human skin (given that 1.5N of force
is required to break the skin): 4.0037 + .4581ln(152.9574) =
6.3080. It would require a Von Frey filament of 6.3080
micrometers to puncture the skin of a human.

The second test was to discover if hyperalgesia is caused by C8
vertebrae spinal cord damage. Hyperalgesia is when a patient
experiences a heightened response to a painful stimulus. While
initial discomfort is understandable, hyperalgesia is present if
the patient reacts far more intensely to the pain than earlier
recorded. To test this, the same rats who received the dorsal
column crush in the first test were placed on an infrared heat
generator. The bottom of the rat's paw was placed on the heater and
heat was applied increasingly until the rat reacted, pulling his
paw away.

Emerson Thacker's research was executed through the use of an
experiment. He manipulated a variable (which in this case was the
rat's C8 vertebrae) in hope of observing a response. He used the
four rats each as an individual sample; however, Thacker's
experiment left room for improvement in three areas. In his
experiment, he samples four male rats, a number too small to create
a statistically significant conclusion for the entire population of
rats. Also, by testing the same rats using both mechanical and
thermal stimulus, one cannot prove that the two variables do not
confound, meaning the data may be biased. However, most strikingly,
Thacker fails to identify a control rat. Without a control to
ensure the predicted or expected results, his experiment lacks
validity and cannot be used to draw a conclusion.

Language Analysis

Astronomy has always been a science of wonder. The stars which
we all see in the moonlit sky are so very distant, yet seem close
enough to grasp in our hands. But how far away are they exactly?
Most normal people will never know. However, Kosan is no normal
person. His super intelligence is eloquently illustrated in "The
Story of Kosan" by Nikhil Desai, in which his aptitude allows him
to solve extraordinary problems. In fact, Kosan "could calculate
the trajectory of the stars all in his head." This mathematical
feat is no simple task.

Astronomers and astrophysicists use a method of comparative
measurements, referred to as a star's parallax shift, to calculate
the trajectory of most stars within a few thousand light years of
Earth. These scientists utilize Earth's orbit, measuring the change
in the star's position in the sky over a six month period. There is
a distinct difference in the perceived location of the star in the
sky with relation to the observer on Earth.

By taking measurements of a star's position from opposite sides
of Earth's orbit, one is left with two equal but unknown distances
to the star. These distances become the legs of an isosceles
triangle. The point where these two legs intersect creates a very
small but significant angle at which the star is the vertex (see
Figure 3). This angle is measured by the observer by comparing the
perceived locations of the star and applying the geometric rule of
alternate interior angles (see Figure 4). Note that these figures
have been exaggerated to emphasize the parallax method and are not
intended to show the relative scale of the distances.

This angle is so small that it is possible and appropriate to
apply the rule of small-angle approximation. This rule states that
when one angle of a right triangle is extremely small, the
hypotenuse and long leg of the triangle are approximately equal.
Thus, the sine of the angle is approximately equal to the angle
itself. This tool is vital to the derivation of the equation for
calculating the distance to a star.

In the case of star trajectory, the sine of the extremely small
angle is equal to 1AU (149,600,000 km) divided by the distance to
the star, sin x = 1AU / d with x representing the
angle in arcseconds and d representing the distance to the
star. However, this equation can be simplified by applying
small-angle approximation, resulting in sin(x) being equal
to x. Rearranged, this equation can be solved for
distance, and the resulting equation is d = 1AU / x. The
final distance is measured
in parsecs (parallax of one arcsecond). One parsec is equal to
about 3.26 light years.

Using this equation, one is able to calculate the distances of
any star in the near astronomic vicinity of Earth. For example, the
most well known star, Polaris (the north star), has a parallax
angle of just .007576 arcseconds and thus has a distance of 132
parsecs, or an astounding 430 light years. Sirius has a parallax
angle of .38 arcseconds and thus a distance of only 2.63 parsecs,
or 8.6 light years. The nearest star to Earth (other than the sun)
is Proxima Centauri, which has a parallax angle of .77619
arcseconds and thus a distance of about 1.28 parsecs, or 4.2 light
years. The distances of parallax-measurable stars can be
graphically represented. Negative x-values, or negative angles, can
be ignored because negative angles are not practical. Furthermore,
x- values larger than .77619 can be ignored, as this would suggest
a star being closer than Proxima Centauri. The resulting graph
illustrates, concisely, the distance to parallax-measurable stars
(see Figure 5).

Art Analysis

For anyone who doesn't have a PhD, the human body can be a bit
overwhelming to understand. Fortunately, we have doctors and
surgeons to help in times of disaster. Just imagine attempting to
put back together a human spine in the correct manner. The average
person wouldn't have any idea where to begin.

Artist Brianna Zahir does an excellent job of portraying the
human spine and the injuries that can occur. She offers the unique
perspective of displaying the individual vertebrae in a circular
manner with the body being supported by this circle. Without the
backbone of the spinal column, there is little hope for the rest of
the body. Similarly, the order of the vertebrae is vital; all 33
individual vertebrae have their assigned location to make the spine
perform at maximum potential. If somebody was to be inflicted with
a spinal injury in which the vertebrae were to become out of order
and had to be reconstructed, one could apply the mathematical
principles of combinatorics to analyze this scenario. We use
Zahir's circular model as our basis for the process.

Combinatorics is a discrete mathematical concept in which
permutations, combinations, and enumerations are performed on sets
of objects to find the number of different possible outcomes.
Essentially, you must decide on whether an object is going to be
used and whether or not the order to the object matters. In our
instance, we are dealing with 33 vertebrae. Because 9 of these
pieces are fused together to form the sacrum and coccyx, there are
essentially 26 interchangeable pieces. This does not indicate that
the 26 vertebrae can merely be placed in any order; there is a
definite order to the spinal column with each vertebra performing a
specific function. We can also add our knowledge that, including
the sacrum and coccyx, there are 5 regions of the spine.

Depending on what information we are given, the number of
options to place the individual vertebra varies significantly. If
we simply know that there are 33 vertebrae, and nothing more, you
would use 33 factorial, or 33! to compute the total available
organizations of vertebrae. 33! = 33x32x31x...x1 = 8.68 x
1036. Adding the knowledge that 9 of the vertebrae
are fused together to make 2 pieces, we have brought the total
number of interchangeable vertebrae down to 26. Using 26! will give
us a smaller but still useless number. If it took 10 seconds to
create each possible arrangement, it would take you 1.28 x
1020 years to try all the possibilities. Going even
further, we have already concluded that there are 5 regions of the
spine. We know how many vertebrae are in each group, so options are
narrowed down considerably. By simply using the factorial of the
number of vertebrae in each group and multiplying by the number of
groups because of the rearrangement of the groups along the spine,
we arrive at the equation N = 7!x5!x12!x1!x1!x5! = 3.48 x
1016. If we already know the location of each
region, this number is reduced by a factor of 5!, down to 2.90
x 1014.

While most combinatorics problems work with situations in a
line, Brianna Zahir's sculpture depicts the spine in a circle.
Figure 4 depicts an example of such an arrangement. In order to
compute the number of possible arrangements of a circular spine, a
much more complex equation is required. We begin by calculating the
number of distinguishable arrangements of 26 objects of 5 distinct
types. There are 26 vertebrae in 5 different regions of the spine.
(Note: in this assumption we are allowing the possibility that all
26 vertebrae could be of the same type, or they could span across
the five types in any way, or anything in between. This is
necessary for the equation in Figure 7 to be applicable). From
these two numbers, we are able to calculate the rest of the
combinations. First, we need to determine the number of divisors of
n. In this case, for n = 26, there are four divisors (1,
2, 13, 26). We then take each of these four divisors and determine
how many positive integers less than the divisor are relatively
prime to the divisor. These are called totatives. The function that
calculates totatives is called the totient function (Φ(di)
). For example, 13 has 12 totatives because the numbers 1-12 are
all relatively prime to 13. For each divisor of n, the totient
function is multiplied with the number of regions (a) to the (n
divided by the divisor) power. Once these two numbers are
multiplied together for each divisor, they are added up using
summation and divided by n (see Figure 7).

When we input our number of vertebrae and the number of regions,
we are able to arrive at the conclusion that there are 5.73 x
1016 possible ways to rearrange 5 regions of 26
vertebrae around a circle. At the same rate of 1 combination per 10
seconds, it would take 18.2 billion years to create every scenario.
With so many options and so little time, it may be best to begin
the reconstruction process as soon as possible.